The restriction “\(\lambda \rho \le r_{K}^{\phi }(a, \eta )\) ” cannot be
avoided since the Lipschitz constant of \(\operatorname{grad}\delta _{K}^{\phi }\)  may explode near
points of \(\operatorname{Cut}^{\phi }(K)\) ; cf. REF . Observe that if \(x \in \mathbf {R}^{n} \operatorname{\sim }\bigl ( \mathop {\mathrm {Clos}}\Sigma ^{\phi }(K) \cup K \bigr )\) , then \( x \) 
has positive distance from \(\operatorname{Cut}^{\phi }(K) \) ; hence,
theorem REF  includes Theorem REF  as a special case;
moreover it is sharp in terms of specifying the set of points where a Lipschitz
condition for \(\operatorname{grad}\delta _{K}^{\phi }\)  holds. There are two main difficulties in
proving REF . The first one arises from the fact that \(\operatorname{Cut}^{\phi }(K) \)  might be dense in \( \mathbf {R}^{n} \operatorname{\sim }K \)  and consequently
it does not seem to be possible to rely on general results for Hamilton-Jacobi
equations as for Theorem REF . The second difficulty arises with working
with a possibly non-Euclidean norm. In fact, if \( \phi \)  is the Euclidean norm
then the proof of Theorem REF  follows rather directly from the
geometric argument originally found by Federer for sets of positive reach in
[1]}, see [2]}. However this argument is not
applicable if \( \phi \)  is not the Euclidean norm, in which case one needs a
considerably more sophisticated approach, based on a careful analysis of the
geometric properties of the \( \phi \) -balls (which occupies the entire section
). In fact this analysis allows to show that the \( \phi \) -nearest point projection \( \xi _{\phi }^{K} \)  onto \( K \) , which is a multivalued
map, satisfies the asserted Lipschitz property; see . Recalling a well known relation between \( \xi _{\phi }^{K} \)  and \( \operatorname{grad}\delta _{K}^{\phi } \)  (see
Lemma  REFREF ),
we obtain the conclusion in REF . We remark that uniform
convexity of the norm for \( n \ge 3 \)  is crucial to obtain the Lipschitz
property in Theorem REF ; see the last section in
[3]}.
